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Inspired by the study of $V$-static manifold about classification, in thisarticle, we apply the recent results obtained by Freitas and Gomes (Compactgradient Einstein-type manifolds with boundary, 2022) to prove the rigidityresults for compact gradient Einstein-type manifolds with nonempty boundary andconstant scalar curvature under some suitable pinching conditions. As a specialcase of gradient Einstein-type manifold, we also give a rigidity result of$(m,\rho)$-quasi-Einstein manifold with boundary.

Compact gradient Einstein-type manifolds with boundary and constant scalar curvature